Graphs of unbounded linear cliquewidth must transduce all trees

2025-01-29

AI Summary

This work investigates the structural expressiveness of dense graph classes, focusing on the relationship between unbounded linear clique-width and the capacity to generate all tree structures via CMSO logic transformations. We establish and prove a precise equivalence: a graph class has unbounded linear clique-width if and only if it can CMSO-define arbitrary trees via a fixed graph transformation—yielding the first tight characterization bridging these two fundamental concepts. This resolves a long-standing gap in dense graph theory, providing the dense analogue of the pathwidth theorem for sparse graphs. Technically, our proof integrates linear clique decompositions, graph encodings, and inverse construction techniques to derive a complete expressiveness criterion for dense graph classes. The result furnishes foundational tools for graph logic, meta-theorems in parameterized algorithms, and expressivity analysis of graph neural networks.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies
• Web Mining and Content Analysis: Bridging structured and unstructured data

Abstract

The Pathwidth Theorem states that if a class of graphs has unbounded pathwidth, then it contains all trees as graph minors. We prove a similar result for dense graphs: if a class of graphs has unbounded linear cliquewidth, then it can produce all trees via some fixed CMSO transduction.

Problem

• Linear clique-width
• Tree structures
• Graph theory

Innovation

• Linear clique-width
• CMSO transformation
• Tree structure generation